Theorem spw 2031

Description: Weak version of the specialization scheme sp 2181. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2181 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2181 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2133 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2181 are spfw 2030 (minimal distinct variable requirements), spnfw 1977 (when 𝑥 is not free in ¬ 𝜑), spvw 1978 (when 𝑥 does not appear in 𝜑), sptruw 1803 (when 𝜑 is true), spfalw 1995 (when 𝜑 is false), and spvv 1994 (where 𝜑 is changed into 𝜓). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)

Hypothesis

Ref	Expression
spw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spw	⊢ (∀𝑥𝜑 → 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem spw

Step	Hyp	Ref	Expression
1		ax-5 1908	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		ax-5 1908	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-5 1908	. 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
4		spw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	1, 2, 3, 4	spfw 2030	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777

This theorem is referenced by:  hba1w  2045  19.8aw  2048  exexw  2049  spaev  2050  ax12w  2131  rspw  3234  reldisj  4459  ralidmw  4514  dtruALT2  5376  dtruOLD  5452  bj-ssblem1  36637  bj-ax12w  36660  eu6w  42663